Anomalous diffusion space

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

It has been assumed in Equation (6) that the tracer can freely access all void space, be it infra- or interparticle. Note that if a barrier to this exchange exists instead, the possibility of the onset of anomalous diffusion should be considered.42 In this case, the molecular displacement does not increase linearly as a function of the echo time, due to the physical threshold, which translates in an apparent reduction of the diffusion coefficients (till vanishing) for increasing A. Thus, the independence of De on the echo time must be controlled in order not to produce erratic experimental values. 

In the previous section we analyzed the random walk of molecules in Euclidean space and found that their mean square displacement is proportional to time, (2.5). Interestingly, this important finding is not true when diffusion is studied in fractals and disordered media. The difference arises from the fact that the nearest-neighbor sites visited by the walker are equivalent in spaces with integer dimensions but are not equivalent in fractals and disordered media. In these media the mean correlations between different steps (UjUk) are not equal to zero, in contrast to what happens in Euclidean space cf. derivation of (2.6). In reality, the anisotropic structure of fractals and disordered media makes the value of each of the correlations u-jui structurally and temporally dependent. In other words, the value of each pair u-ju-i-- depends on where the walker is at the successive times j and k, and the Brownian path on a fractal may be a fractal of a fractal [9]. Since the correlations u.juk do not average out, the final important result is (UjUk) / 0, which is the underlying cause of anomalous diffusion. In reality, the mean square displacement does not increase linearly with time in anomalous diffusion and (2.5) is no longer exact. 

The type of the random walk (recurrent or nonrecurrent) determines the minimum value of the two terms in the brackets of the previous equation. If the walker does not visit the same sites (nonrecurrent) then dw = 2df/ds. If the walk is of recurrent type then the walker visits the same sites again and again and therefore the walker covers the available space (space-filling walk). Consequently, the meaning of dw coincides with df (dw = df). The mean square displacement in anomalous diffusion follows the pattern... 

We will discuss this state in relation to the recent approaches of the anomalous diffusion theory [31]. It is well known [226-230] that by virtue of the divergent form of Poisson brackets (95) the evolution of the distribution function pip,q t) can be regarded as the flow of a fluid in phase space. Thus the Liouville equation (93) is analogous to the continuity equation for a fluid... 

Equation (5) is a partial differential equation of infinite order, and cannot be solved in general. Since all properties of glass vary slowly in space and time, the left-hand side of Eq. (5) can be truncated, the motion of holes in response to molecular fluctuations is then treated as an anomalous diffusion process [12]... 

In this chapter we considered only a small Hamiltonian system whose Poincare map is the standard map defined on the unit square. It is interesting to consider Hamiltonian systems in a large phase space in which diffusion appears. Specifically, we are interested how the accelerator mode, which causes the anomalous diffusion in the standard map, affects the breaking of the adiabatic invariant. We will continue this study in a forthcoming article [21]. 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

The crossover from anomalous to normal diffusion determines the time when the anomalous diffusion finishes. However, it is not clearly pointed out when the anomalous diffusion starts, and hence the study of the relation between the relaxation process and anomalous diffusion is still not complete. Moreover, in Ref. 18, the numerical calculations were performed by using only one type of initial condition—that is, the waterbag initial condition giving a /-exponential distribution [15]—but different types of initial condition may change the conclusion. For instance, in the one-dimensional self-gravitating sheet model, the waterbag initial condition gives a power-type spatial correlation, but a thin width of initial distribution on p-space breaks the power law [29]. 

In a Hamiltonian system having mean field interaction, referred to as the HMF model, we have investigated two features that must reflect self-similar hierarchy of phase space power-type distribution and anomalous diffusion. They have been reported in the same model for one type of initial condition, and we used a different type of initial condition to check generality. 

If Eq. (14) holds then anomalous diffusion may appear only for D = 0 and very strong Lagrangian velocity correlations. The latter condition can be realized— for example, in time periodic velocity fields where the Lagrangian phase space has a complicated self-similar structure of islands and cantori [30]. Here superdiffusion is due to the almost trapping, for arbitrarily long time, of the ballistic trajectories close to the cantori, which are organized in complicated selfsimilar structures. 

Dielectric loss %"((d) and absorption dry" ( ) spectra for various values of ot and (3/ are shown in Figs. 26-29. The Cole-Cole plot y"(o>) versus yff(d) is presented in Fig. 20. It is apparent that the half-width and the shape of dielectric spectra strongly depend on both ot (which in the present context pertains to anomalous diffusion in velocity space) and (3 (which characterizes the effects of molecular inertia). In the high damping limit ((3 1) and for a > 1 corres-... 

Time and Space Resolution Required to Observe Anomalous Diffusion of a Single Molecule in Biological Tissues... 

This aspect of diffusion in inhomogeneous space is called anomalous diffusion [13-18] and ordinal diffusion which can be expressed as a Brownian motion is called normal diffusion or Euclid diffusion [14]. The definition of... 

In practice, poor time and space resolution lead to a wrong evaluation of the diffusion coefficient in a case including anomalous diffusion. One typical example is the use of video cameras with different frame speeds. With a vector model as shown in Figure 33.4, the results of two different detections for the same Brownian motion... 

On the other hand, the effectiveness of the signaling reactions also depends on the diffusion coefficient, as shown in Eq. (33.11). Although other parameters in Eq. (33.11) (rs, rA, and nT) are not variable as determined for each reaction (33.10), only the diffusion coefficients (Ds and DA) can be controlled by the existence of the surrounding media. Moreover, as mentioned in the previous section, the diffusion coefficient of anomalous diffusion depends on the diffusion time and the dimensions of the reaction space. In such a situation, the diffusion coefficient observed by one method (e.g., FCS, FRAP) is only a local value, depending on the time constant and the spatial size of a proper experiment. As mentioned for Figure 33.4 in the beginning of this section, the size of the reaction volume for signaling reaction is of the order of pL-fL and measurement of the diffusion coefficient in such a microspace is important. 

For small molecules, the decrease in D occurs at I slightly larger than the mesh size. The small molecule is gradually decelerated passing through several numbers of mesh units. This mechanism is frequently referred to as the Ant in the Labyrinth mechanism in percolation theory [33, 34]. Application of percolation to anomalous diffusion in the mesh space suggests the possible appearance of low-dimensional (tube-like) transportation inside the space. This phenomenon will be detected by another type of anomalous behavior in DDDC or TDDC. 

The results of the study in HA solution suggest a simple but secure strategy to control the reaction by anomalous diffusion occurring in biological systems. The simple polymer solution used here is not only a model for ECM but also one to be extended to other biological space such as cytoplasm and membranes. 

These values are consistent with our previous results concerning anomalous diffusion in HA solution in which the location of the transient anomalous diffusion area is 10-100 nm. The existence of HA never disturbs a reaction space smaller than the 10-100 nm scale where the observed value of D is unchanged from the value without HA (D0). If the signaling molecules are provided within this small volume, in... 

New fluorescence correlation spectroscopy (FCS) suitable for the observation of anomalous diffusion in polymer solution time and space dependences of diffusion coefficients. 

Although anomalous diffusion is expected in fractal pore systems, the presence of anomalous diffusion does not prove that the porous media is fractal. A heterogeneity along transport pathways may result in an anomalous transport regardless of the presence or the absence of self-similarity of the pore space (Beven et al., 1993). The physical interpretation of Levy motions does not presume the presence of fractal scaling in the porous media in which the motions occur (Klafter et al, 1990). The applicability of the FADE may be closely related to the distribution of pore-water velocities. In saturated media, the presence of heavy-tailed distributions of the hydraulic conductivity directly implies the validity of the FADE (Meer-schaert et al., 1999 Benson et al., 1999). The heavy-tailed hydraulic conductivity distributions were found in geologic media (Painter, 1996 Benson et al., 1999). Heavy-tailed velocity distributions can also be expected in unsaturated and structured soils, and therefore the FADE may be a useful model in these conditions. 

Recent advances in percolation theory and fractal geometry have demonstrated that Dc is not a constant when diffusion occurs as a result of fractional Brownian motion, i.e., anomalous diffusion (Sahimi, 1993). The time-dependent diffusion coefficient, D(t), for anomalous diffusion in two-dimensional free space is given by (Mandelbrot Van Ness, 1968),... 

The anomalous diffusivity described by Eq. [13] is due entirely to the fractal nature of the diffusing particle s trajectory in free space. In fractal and multifractal porous media, the diffusing particle s trajectory is further constrained by the geometry of the pore space (Cushman, 1991 Giona et al., 1996 Lovejoy et al., 1998). As a result, when fractional Brownian motion occurs in a two-dimensional fractal porous medium, De becomes scale-dependent, as described by the following equation (Orbach, 1986 Crawford et al., 1993),... 

Anomalous diffusion of a continuous concentration field can be modelled in terms of fractional differential equations. To see how they arise we can write Eq. (2.9) for normal diffusion in terms of the spatial Fourier transform of the concentration field C(k, t). This can be easily done under periodic boundary conditions or in unbounded space as... 

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333], The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes standard diffusion, tfansport with inertia, and anomalous diffusion. 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

Anomalous diffusion describes a transport process analogous to transient diffusion which does not follow the classical model based on Pick s law. The dependence with the frequency of the electrical impedance featuring this transfer, which is normally 1/2 for the Fickian diffusion in infinite space, is different from this value. The same is true for the degree... 

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